Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2011, Article ID 635823, 21 pages
doi:10.1155/2011/635823
Research Article
Optimal Bounded Control for Stationary Response
of Strongly Nonlinear Oscillators under Combined
Harmonic and Wide-Band Noise Excitations
Yongjun Wu,1 Changshui Feng,2 and Ronghua Huan3
1
Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China
Institute of Mechatronic Engineering, Hangzhou Dianzi University, Hangzhou 310018, China
3
Department of Mechanics, Zhejiang University, Hangzhou 310027, China
2
Correspondence should be addressed to Yongjun Wu, yongjun.wu@yahoo.com
Received 29 March 2011; Revised 8 July 2011; Accepted 25 July 2011
Academic Editor: Angelo Luongo
Copyright q 2011 Yongjun Wu et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We study the stochastic optimal bounded control for minimizing the stationary response of
strongly nonlinear oscillators under combined harmonic and wide-band noise excitations. The
stochastic averaging method and the dynamical programming principle are combined to obtain the
fully averaged Itô stochastic differential equations which describe the original controlled strongly
nonlinear system approximately. The stationary joint probability density of the amplitude and
phase difference of the optimally controlled systems is obtained from solving the corresponding
reduced Fokker-Planck-Kolmogorov FPK equation. An example is given to illustrate the
proposed procedure, and the theoretical results are verified by Monte Carlo simulation.
1. Introduction
The well-known tool to solve the problem of stochastic optimal control is the dynamical
programming principle, which was proposed by Bellman 1. According to this principle,
a stochastic optimal control problem may be transformed into the problem of finding a
solution to the so-called Hamilton-Jacobi-Bellman HJB equation. However, in most cases,
the HJB equation cannot be solved analytically 2, especially in the multidimensional case.
A powerful technique to solve stochastic optimal control problems is to combine the socalled stochastic averaging method 3 with Bellman’s dynamic programming principle. The
idea is to replace the original stochastic system by the averaged one and then to apply
the dynamic programming principle to the averaged system. It has been justified that the
optimal control for the averaged system is nearly optimal for the original system 4. This
combination strategy has two notable advantages. Firstly, the dimension of the averaged
2
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system can be reduced remarkably, and the corresponding HJB equation is low-dimensional.
Secondly, the diffusion matrix of the original stochastic system is usually singular, while the
averaged one is usually nonsingular. This unique characteristic enables the HJB equation for
the averaged system to have classical rather than viscous solution. In recent years, a notable
nonlinear stochastic optimal control strategy has been proposed for the control of quasiHamiltonian systems under random excitations by Zhu based on the stochastic averaging
method for quasi-Hamiltonian systems and the stochastic dynamical programming principle
3. Examples have shown that this strategy is very effective and efficient.
In practice, the excitations of dynamical systems can be classified as either deterministic or random. The random excitation is often modeled as Gaussian white noise, wideband colored noise, or bounded noise. On the other hand, the magnitudes of the control
forces are usually limited due to the saturation in actuators; that is, the control forces
are bounded. The optimal bounded control of linear or nonlinear systems under random
excitations has been studied by many researchers 5–8. In all these studies, the excitations of
the systems are random excitations alone. However, many physical or mechanical systems
are subjected to both random and deterministic harmonic excitations. For example, such
combined excitations arise in the study of stochastic resonance 9 or uncoupled flapping
motion of rotor blades of a helicopter in forward flight under the effect of atmospheric
turbulence 10. Due to the existence of the deterministic harmonic excitation, the response
of the stochastic system is not time homogeneous and the stationary behavior is not easy
to capture. Stochastic averaging method is such a method that can approximate the original
system by time-homogeneous stochastic processes. By using stochastic averaging method,
many researchers have studied the linear or strongly nonlinear systems under combined
harmonic and wide-band random excitations 11–16. On the other hand, optimal bounded
control of a linear or nonlinear oscillator subject to combined harmonic and Gaussian white
noise excitations has been studied 17–19. However, Gaussian white noise is an ideal model.
In most cases, the random excitation should be modeled as wide-band colored noise. So far,
little work has been done on the optimal bounded control of a strongly nonlinear oscillator
subject to combined harmonic and wide-band noise excitations 20.
In the present paper, a procedure for designing optimal bounded control to minimize
the stationary response of a strongly nonlinear oscillator under combined harmonic and
wide-band colored noise excitations is proposed. In Section 2, based on the stochastic
averaging method, the equation of motion of a weakly controlled strongly nonlinear oscillator
under combined harmonic and wide-band noise excitations is reduced to partially averaged
Itô stochastic differential equations. In Section 3, a dynamical programming equation for
the control problem of minimizing response of the system is formulated from the partially
averaged Itô stochastic differential equations by applying the dynamical programming
principle. The optimal control law is determined by the dynamical programming equation
and the control constraint. In Section 4, the reduced FPK equation governing the stationary
joint probability density of the amplitude and phase difference of the optimally controlled
system is established. In Section 5, a nonlinearly damped Duffing oscillator under combined
harmonic and wide-band noise excitations is taken as an example to illustrate the application
of the proposed procedure. All theoretical results are verified by Monte Carlo simulation.
2. Partially Averaged Itô Stochastic Differential Equations
Consider a weakly controlled strongly nonlinear oscillator subject to weak linear or nonlinear
damping and weak external or parametric excitation by a combination of harmonic functions
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3
and wide-band colored noises. The equation of motion of the system has the following
form
Ẍ gX εf X, Ẋ, Ωt εu X, Ẋ ε1/2 hk X, Ẋ ξk t,
k 1, 2, . . . , r,
2.1
where the term gX is stiffness, which is assumed as an arbitrary nonlinear function of X. ε
is a small parameter; εfX, Ẋ, Ωt accounts for light damping and weak harmonic excitation
with frequency Ω; ε1/2 hk ξk t represent weak external and/or parametric random excitations;
ξk t are wide-band stationary and ergodic noises with zero mean and correlation functions
Rkl τ or spectral densities Skl ω; εu denotes a weakly feedback control force. The repeated
subscripts indicate summation.
When u 0, 2.1 describes a large class of physical or structural systems. Under the
conditions specified by Xu and Cheung 21, the solutions of system 2.1 can be expressed
as the following form:
Xt A cos Φt BA,
Ẋt −AνA, Φ sin Φt,
2.2
where
Φt μt Θt,
dμ
νA, Φ dt
2.3
2P A B − P A cos Φ B
,
A2 sin2 Φ
P x x
g y dy.
2.4
2.5
0
A, Φ, μ, Θ, and ν are all stochastic processes. P x is the potential energy of the
system 2.1. The functions cos Φt and sin Φt are called generalized harmonic functions.
Obviously νA, Φ is the instantaneous frequency of the system 2.1.
When ε 0, 2.1 degenerates to the following nonlinear conservative oscillator:
Ẍ gX 0.
2.6
The average frequency of the conservative oscillator 2.6 can be obtained by the following
formula:
ωA 2π
0
2π
dΦ/νA, Φ
.
2.7
Then the following approximate relation exists:
Φt ≈ ωAt Θt.
2.8
4
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Treating 2.2 as generalized Van der Pol transformation from X, Ẋ to A, Θ , one
can obtain the following equations for A and Θ:
dA
1
2
εF1 A, Φ, Ωt εF1 A, Φ, u ε1/2 U1k A, Φξk t,
dt
2.9
dΘ
1
2
εF2 A, Φ, Ωt εF2 A, Φ, u ε1/2 U2k A, Φξk t,
dt
where
1
F1 2
F1 1
F2 −A
fA cos Φ B, −AνA, Φ sin Φ, ΩtνA, Φ sin Φ,
gA B1 h
−Au
νA, Φ sin Φ,
gA B1 h
−1
fA cos Φ B, −AνA, Φ sin Φ, ΩtνA, Φcos Φ h,
gA B1 h
F2 −u
νA, Φcos Φ h,
gA B1 h
U1k −A
hk A cos Φ B, −AνA, Φ sin ΦνA, Φ sin Φ,
gA B1 h
U2k −1
hk A cos Φ B, −AνA, Φ sin ΦνA, Φcos Φ h,
gA B1 h
2
h
2.10
dB g−A B gA B
.
dA g−A B − gA B
According to the Stratonovich-Khasminskii limit theorem 22, 23, A and Θ converge
weakly to 2-dimensional diffusive Markov processes in a time interval of ε−1 order as ε → 0,
which can be represented by the following Itô stochastic differential equations:
2
dA ε S1 A, Φ, Ωt F1 A, Φ, u dt ε1/2 G1k A, ΦdBk t,
2
dΘ ε S2 A, Φ, Ωt F2 A, Φ, u dt ε1/2 G2k A, ΦdBk t,
2.11
where Bk t are independent unit Wiener processes,
1
Si F i
0 −∞
bij εGik Gjk ∂Uik ∂Uik ·
U
R
·
U
R
dτ,
τ
τ
|
|
1l tτ kl
2l tτ kl
∂A t
∂Φ t
∞ −∞
Uik |t · Ujl tτ Rkl τ dτ,
i, j 1, 2, k, l 1, . . . , r.
2.12
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5
System 2.1 has harmonic excitation and two cases can be classified: resonant case
and nonresonant case. In the nonresonant case, the harmonic excitation has no effect on the
first approximation of the response. Thus, we are interested in the resonant case, namely,
m
Ω
εσ,
ωA
n
2.13
where m and n are relatively prime positive small integers and εσ is a small detuning
parameter. In this case, multiplying 2.13 by t and utilizing the approximate relation 2.8
yield
Ωt m
m
Φ εσμ − Θ.
n
n
2.14
Introduce a new angle variable Γ such that
Γ εσμ −
m
Θ
n
2.15
which is a measure of the phase difference between the response and the harmonic excitation.
Then 2.14 can be rewritten as
Ωt m
Φ Γ.
n
2.16
Using the Itô differential formula, one can obtain the following Itô stochastic differential equations for A, Γ, and Φ:
2
dA ε S1 A, Φ, Γ F1 A, Φ, u dt ε1/2 G1k A, ΦdBk t,
m
m
2
S2 A, Φ, Γ F2 A, Φ, u dt − ε1/2 G2k A, ΦdBk t,
dΓ ε σωA −
n
n
2
dΦ ωA ε S2 A, Φ, Γ F2 A, Φ, u dt ε1/2 G2k A, ΦdBk t.
2.17
Obviously, A and Γ are slowly varying processes, while Φ is a rapidly varying process.
Averaging the drift and diffusion coefficients in 2.17 with respect to Φ yields the following
partially averaged Itô stochastic differential equations:
1
2
dt ε1/2 σ 1r AdBr t,
dA ε m1 A, Γ F1 A, Φ, u
Φ
m 2
1
F2 A, Φ, u
dΓ ε m2 A, Γ −
dt ε1/2 σ 2r AdBr t,
Φ
n
2.18
6
Mathematical Problems in Engineering
where
1
m1 S1 A, Φ, ΓΦ ,
1
m2 σωA −
m
S2 A, Φ, ΓΦ ,
n
b11 A εσ 1r σ 1r b11 A, ΦΦ ,
m2
b22 A, ΦΦ ,
n2
m
b12 A b21 A εσ 1r σ 2r b12 A, ΦΦ ,
n
2π
1
•dΦ.
•Φ 2π 0
b22 A εσ 2r σ 2r 2.19
Herein, •Φ denotes the averaging with respect to Φ from 0 to 2π · bij are diffusion
coefficients.
Note that there are two procedures of averaging in this section. One is stochastic
averaging, and the other is deterministic time averaging. The procedures for obtaining Si
1
and bij in 2.12 are called stochastic averaging. The procedures for obtaining mi and
bij are called deterministic time averaging. To complete averaging and obtain the explicit
1
expressions of Si and bij , the functions Fi and Uik in 2.12 are expanded into Fourier series
with respect to Φ and the approximate relation 2.8 is used.
3. Dynamical Programming Equation and Optimal Control Law
For mechanical or structural systems, At is the response amplitude and Γt is the phase
difference between the response and the harmonic excitation. They represent the averaged
state of system 2.1. Usually, only the response amplitude is concerned, so it is meaningful
to control At in a semi-infinite or finite time interval. Consider the ergodic control problem
of system 2.18 in a semi-infinite time interval with the following performance index:
1
J lim
T →∞T
T
fAtdt.
3.1
0
Herein, f is a cost function. Based on the stochastic dynamical programming principle 24,
the following simplified dynamical programming equation can be established from the first
equation of 2.18:
η min fA u
1
m1
1
d2 V
,
b11
dA 2
dA2
dV
2
F1
Φ
3.2
where V A is called value function; minu∈U • denotes the minimum value of • with
respect to u; η is optimal performance index; u ∈ U indicates the control constraint.
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7
The optimal control law can be determined from minimizing the right-hand side of
3.2 with respect to u under control constraint. Suppose that the control constraint is of the
form
|uA, Φ| ≤ u0 ,
3.3
where u0 is a positive constant representing the maximum control force. The control u enters
the performance index 3.2 only through the term
2
F1
dV u
dV
−
AνA, Φ sin Φ
.
Φ dA
gA B1 h
Φ dA
3.4
This expression attains its minimum value for u ±u0 , depending on the sign of the
coefficient of u in 3.4. So the optimal control is
u∗ u0 sgn
AνA, Φ sin Φ dV
·
,
gA B1 h dA
3.5
where sgn denotes sign function.
It is reasonable to assume that the cost function f is a monotonously increasing
function of At because the controller must do more work to suppress larger amplitudes
At. Then dV A/dA is positive. Under the specified conditions 21, gA B and 1 h
are both positive. Equation 3.5 is reduced to
u∗ −u0 sgn−AνA, Φ sin Φ −u0 sgn Ẋt .
3.6
Equation 3.6 implies that the optimal control is a bang-bang control. u∗ has a constant
magnitude u0 . It is in the opposite direction of Ẋt and changes its direction at Ẋt 0.
For the optimal bounded control of system 2.18 in a finite time interval, the following
performance index is taken:
JE
t
f
,
fAtdt χ A tf
3.7
0
where tf is the final control time and χ is the final cost function. E• denotes the expectation
operator. The following simplified dynamical programming equation can be established from
the first equation of 2.18:
∂V 1
∂2 V
∂V
1
2
,
−min fA m1 F1
b11
u
Φ ∂A
∂t
2
∂A2
3.8
where V V A, t is the value function. Equation 3.8 is subject to the following final time
condition:
V A, tf χ A tf .
3.9
8
Mathematical Problems in Engineering
It is obvious that the same optimal control law as that in 3.6 can be derived if the
control constraint is of the form of 3.3.
Note that the optimal control for the averaged system 2.18 is nearly optimal for the
original system 2.1. For simplicity, here it is called optimal control for both original and
averaged systems.
4. Stationary Response of Optimally Controlled System
2
Inserting u∗ from 3.6 into 2.18 to replace u and averaging Fi , the following fully
averaged Itô stochastic differential equations for A and Γ can be obtained:
dA εm1 A, Γdt ε1/2 σ 1k AdBk t,
4.1
dΓ εm2 A, Γdt ε1/2 σ 2k AdBk t,
where
m1 A, Γ m2 A, Γ 1
m1
1
m2
−u∗ AνA, Φ sin Φ
gA B1 h
,
Φ
m u∗ νA, Φcos Φ h
.
n
gA B1 h
Φ
4.2
The reduced FPK equation for the optimally controlled system is of the following form
0−
∂
1 ∂2 1 ∂2 ∂ ∂2 m1 p −
m2 p b11 p b12 p b22 p ,
2
2
∂a
∂γ
2 ∂a
∂a∂γ
2 ∂γ
4.3
where p pa, γ is the stationary joint probability density of the amplitude A and the phase
difference Γ. Since p is a periodic function of γ, it satisfies the following periodic boundary
condition with respect to γ:
p a, γ p a, γ 2π .
4.4
The boundary condition with respect to a 0 is
p finite at a 0
4.5
which implies that a 0 is a reflecting boundary. The other boundary condition is
p,
∂p
−→ 0
∂a
as a −→ ∞.
4.6
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9
In addition to the boundary conditions, the stationary joint probability density pa, γ
satisfies the following normalization condition:
2π ∞
0
p a, γ da dγ 1.
4.7
0
Usually, the partial differential equation 4.3 can be solved only numerically.
5. Example
To illustrate the proposed strategy in the previous sections, take the following controlled
nonlinearly damped Duffing oscillator as an example. Duffing oscillator is a typical model in
nonlinear analysis. The equation of motion of the system is of the form
Ẍ β1 β2 X 2 Ẋ ω02 X αX 3 E cos Ωt ξ1 t Xξ2 t u,
5.1
where β1 , β2 , ω0 , α, E, and Ω are positive constants; u is the feedback control with the
constraint defined by 3.3; ξk t k 1, 2 are independent stationary and ergodic wideband noises with zero mean and rational spectral densities
Si ω 1
Di
,
π ω2 ωi2
i 1, 2.
5.2
ξi t can be regarded as the output of the following first-order linear filter:
ξ̇i ωi ξi Wi t,
i 1, 2,
5.3
where Wi t are Gaussian white noises in the sense of Stratonovich with intensities 2Di . Note
that the maximum value of Si ω is Di /πωi2 . It is assumed that β1 , Di /πωi2 , and E are all
small.
For the system 5.1, the instantaneous frequency defined by 2.4 has the following
form:
νA, Φ ω02 3αA2
4
1/2
1 λ cos 2Φ
,
5.4
αA2
λ 2
.
4 ω0 3αA2 /4
νA, Φ can be approximated by the following finite sum with a relative error less than 0.03%:
νA, Φ b0 A b2 A cos 2Φ b4 A cos 4Φ b6 A cos 6Φ,
5.5
10
Mathematical Problems in Engineering
where
b0 A b2 A b4 A b6 A ω02 3αA2
4
ω02 3αA2
4
ω02 3αA2
4
ω02 3αA2
4
1/2 λ2
1−
16
1/2 ,
λ 3λ3
2
64
1/2 λ2
−
16
1/2 λ3
64
,
5.6
,
.
Then the averaged frequency ωA of the system 5.1 can be approximated by b0 A.
In the case of primary external resonance,
Ω
1 εσ.
ωA
5.7
Introduce the new angel variable defined by 2.15, and complete the procedures
shown in Sections 2–4 we obtain the following fully averaged Itô stochastic differential
equations for A and Γ:
b11 AdB1 t,
dΓ m2 A, Γdt b22 AdB2 t,
dA m1 A, Γdt 5.8
where mi and bii i 1, 2 are drift and diffusion coefficients, respectively. mi and bii are given
in the appendix.
The stationary joint probability density pa, γ of the optimally controlled system 5.1
is governed by the following reduced FPK equation:
0−
1 ∂2 ∂
1 ∂2 ∂ p
p
.
m1 p −
m2 p b
b
11
22
∂a
∂γ
2 ∂a2
2 ∂γ 2
5.9
Solving the FPK equation 5.9 by finite difference method under the conditions 4.4–
4.7, the stationary joint probability density of the amplitude and the phase difference of
the optimally controlled system 5.1 can be obtained. Furthermore, the stationary mean
amplitude of the optimally controlled system 5.1 can be obtained as follows:
Ea ∞ 2π
0
0
ap a, γ da dγ.
5.10
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11
2.5
2
1.5
1
ξ1 (t)
0.5
0
−0.5
−1
−1.5
−2
−2.5
0
50
100
150
200
t
250
300
350
400
Figure 1: Sample function of wide-band noise ξ1 t · D1 10, ω1 30.
0.1
0.05
u
0
−0.05
−0.1
−0.15
0
20
40
60
80
100 120 140 160 180 200
t
Figure 2: Sample function of the control force u, u0 0.1.
To check the accuracy of the proposed method, Monte Carlo digital simulation of
the original system 5.1 is performed. The sample functions of independent wide-band
noise ξi t were generated by inputting Gaussian white noises to the linear filter 5.3. The
response of system 5.1 was obtained numerically by using the fourth-order Runge-Kutta
method with time step 0.02. The long-time solution after 1500,000 steps was regarded as the
stationary ergodic response. 100 samples are used. For every sample, the amplitude A and
angle variable Γ are calculated from step 1500,001 to step 2000000 to obtain the statistical
probability density of pa, γ. Figures 1 and 2 show the typical sample function of wideband noise ξ1 t and control force u, respectively. Figure 3 shows pa, γ of the optimal
controlled system 5.1. It is seen that the theoretical result agrees very well with that from
digital simulation. Figures 4a and 4b show the sample functions of displacement and
velocity of the system 5.1, respectively. It is obvious that the bounded control can reduce
the displacement and velocity. Also, the amplitude of the original system 5.1 is reduced by
control, which is verified by Figure 5.
Mathematical Problems in Engineering
2
1.6
1.2
0.8
0.4
0
p(a, γ)
2
1.6
1.2
0.8
0.4
0
2
1.5
1
a
2
0.5
3
4
5
p(a, γ)
12
6
γ
1
0 0
a
1.6
1.2
0.8
0.4
0
2
1.6
1.2
0.8
0.4
0
p(a, γ)
p(a, γ)
2
2
1.5
a
1
2
4
3 γ
5
6
1
0.5
0 0
b
0.8
2.5
0.7
0.6
0.5
1.5
p(γ)
p(a)
2
1
0.3
0.2
0.5
0
0.4
0.1
0
0.5
1
c
a
1.5
2
2.5
0
0
1
2
3
γ
4
5
6
d
Figure 3: Stationary joint probability density pa, γ of the optimally controlled system 5.1 in primary
external resonance case. ω0 1.0, Ω 1.1, α 1.0, E 0.3, βi 0.05, ωi 30, D1 10, D2 5, u0 0.1
i 1, 2. a Theoretical result; b Monte Carlo simulation of the original system 5.1; c stationary
marginal probability density pa; d stationary marginal probability density pγ. — theoretical results;
• results from Monte Carlo simulation.
Note that the system 5.1 is strongly nonlinear. Increasing the nonlinearity coefficients
β1 or α in 5.1, one can see that the agreement between theoretical results and Monte
Carlo digital simulation is still acceptable see Figures 6 and 7. This demonstrates that the
proposed method is powerful to deal with strongly nonlinear problems, even though the
nonlinearity is extremely strong.
Mathematical Problems in Engineering
13
2
1.5
1
0.5
0
X
−0.5
−1
−1.5
−2
30,000
30,100
30,200
t
30,300
30,400
30,300
30,400
a
4
3
2
1
Ẋ 0
−1
−2
−3
−4
30,000
30,100
30,200
t
b
Figure 4: Sample functions of displacement and velocity of the system 5.1, u0 0.4. Other parameters
are the same as those in Figure 3. Red line: with control; blue line: without control.
0.9
α=1
0.8
0.7
E[a]
0.6
0.5
0.4
α=3
0.3
0.2
0.1
0
0.05
0.1
0.15
0.2
u0
Figure 5: Mean amplitude of the response of the optimally controlled system 5.1 in primary external
resonance. The parameters are the same as those in Figure 3 except that u0 is a variable. — theoretical
results; • results from Monte Carlo simulation.
14
Mathematical Problems in Engineering
7
β2 = 100
6
p(a)
5
4
β2 = 6
3
2
β2 = 0.1
1
0
0
0.5
1
1.5
2
2.5
a
Figure 6: Stationary marginal probability density pa of the optimally controlled system 5.1 in primary
external resonance case. Other parameters are the same as those in Figure 3. — theoretical results; • results from Monte Carlo simulation.
7
6
α = 100
p(a)
5
α = 15
4
3
α=5
2
1
0
0
0.2
0.4
0.6
0.8
1
a
Figure 7: Stationary marginal probability density pa of the optimally controlled system 5.1 in primary
external resonance case. Other parameters are the same as those in Figure 3. — theoretical results; • results from Monte Carlo simulation.
6. Conclusions
In the present paper, a combination procedure of the stochastic averaging method and
Bellman’s dynamic programming for designing the optimal bounded control to minimize
the response of strongly nonlinear systems under combined harmonic and wide-band noise
excitations has been proposed. The procedure consists of applying the stochastic averaging
method for weakly controlled strongly nonlinear systems under combined harmonic and
wide-band noise excitations, establishing the dynamical programming equation for the
control problem of minimizing the response based on the partially averaged Itô stochastic
differential equations and the dynamical programming principle, determining the optimal
control from the dynamical programming equation and the control constraint without
solving the dynamical programming equation. Then the stationary joint probability density
Mathematical Problems in Engineering
15
and mean amplitude of the optimally controlled averaged system are obtained from solving
the reduced FPK equation associated with the fully averaged Itô stochastic differential
equations. A nonlinearly damped Duffing oscillator with hardening stiffness has been taken
as an example to illustrate the application of the proposed procedure. The comparison
between the theoretical results and those from Monte Carlo simulation shows that the
proposed procedure works quite well even though the nonlinearity is extremely strong. The
results show that the response amplitude of the system can be reduced remarkably by the
feedback control.
The advantages of the proposed method are obvious. Note that; in the example, the
wide-band noise is generated by a first-order linear filter. In principle, one could apply the
T
stochastic dynamical programming method to the extended system X, Ẋ, ξ1 , ξ2 to study
the optimal control problem. However, the corresponding HJB equation is 5-dimensional
or 4 dimensional. The corresponding reduced FPK equation is 4 dimensional, which is
very difficult to solve. After stochastic averaging, the original system is represented by
two-dimensional time-homogeneous diffusion Markov processes of amplitude and phase
difference with nondegenerate diffusion matrix. The dynamical programming equation
derived from the averaged equations is two dimensional or one dimensional. The
corresponding reduced FPK equation is two dimensional, which is easy to solve. The other
advantage of the proposed procedure is that it is not necessary to solve the dynamical
programming equation for obtaining the optimal control law. Furthermore, the proposed
method can be extended to multi-degrees-of-freedom MDOF systems easily. This will be
our future work.
Appendix
The drift coefficients in 5.8 are as follows:
m1 H 1 A F 10 A, Γ 2u0 −105b0 A 35b2 A 7b4 A 3b6 A
,
105π ω02 αA2
m2 H 2 A Ω − b0 A F 10 A, Γ E cos Γ2b0 A b2 A
,
2
4A αA ω02
E sin Γ2b0 A − b2 A
2
4 αA ω02
A β1 16ω02 10αA2 A2 β2 4ω02 3αA2
−
,
2
32 αA ω02
H 1 A m11 m12 m13 m14 ,
H 2 A m21 m22 m23 m24 ,
m11 m111 S1 ωA m113 S1 3ωA m115 S1 5ωA m117 S1 7ωA,
m13 m131 S1 ωA m133 S1 3ωA m135 S1 5ωA m137 S1 7ωA,
m12 m122 S2 2ωA m124 S2 4ωA m126 S2 6ωA m128 S2 8ωA,
16
Mathematical Problems in Engineering
m14 m142 S2 2ωA m144 S2 4ωA m146 S2 6ωA m148 S2 8ωA,
m21 m211 I1 ωA m213 I1 3ωA m215 I1 5ωA m217 I1 7ωA,
m23 m231 I1 ωA m233 I1 3ωA m235 I1 5ωA m237 I1 7ωA,
m22 m222 I2 2ωA m224 I2 4ωA m226 I2 6ωA m228 I2 8ωA,
m24 m242 I2 2ωA m244 I2 4ωA m246 I2 6ωA m248 I2 8ωA,
Ii ω ω
Di
2
ωi ω ωi2
i 1, 2,
m111 πb2 A − 2b0 A
2αA2b0 A − b2 A A2 α ω02 db2 A/dA − 2db0 A/dA
×
,
3 8 A2 α ω02
m113 πb2 A − b4 A
2αAb4 A − b2 A A2 α ω02 db2 A/dA − db4 A/dA
,
×
3 8 A2 α ω02
m115 πb4 A − b6 A
2αAb6 A − b4 A A2 α ω02 db4 A/dA − db6 A/dA
,
×
3 8 A2 α ω02
m117
πb6 A −2αAb6 A A2 α ω02 db6 A/dA
,
3 8 A2 α ω02
m122 πA2b0 A − b4 A
2b0 A−b4 A A2 α − ω02 − A A2 α ω02 2db0 A/dA − db4 A/dA
×
,
3 32 A2 α ω02
m124 πAb2 A − b6 A
b6 A−b2 A A2 α−ω02 A A2 αω02 db2 A/dA−db6 A/dA
×
,
3 32 A2 α ω02
m126
πAb4 A b4 A A2 α − ω02 A A2 α ω02 db4 A/dA
,
3 32 A2 α ω02
m128
πAb6 A b6 A A2 α − ω02 A A2 α ω02 db6 A/dA
,
3 32 A2 α ω02
Mathematical Problems in Engineering
−π b22 A − 4b02 A
m131 2 ,
8A A2 α ω02
m133
−3π b42 A − b22 A
2 ,
8A A2 α ω02
m135
−5π b62 A − b42 A
2 ,
8A A2 α ω02
7πb2 A
m137 6
2 ,
8A A2 α ω02
m142 πA2b0 A − b4 A2b0 A 2b2 A b4 A
,
2 16 A2 α ω02
m144 πAb2 A − b6 Ab2 A 2b4 A b6 A
,
2 8 A2 α ω02
m146 3πAb4 Ab4 A 2b6 A
,
2 16 A2 α ω02
πAb62 A
m148 2 ,
4 A2 α ω02
2b0 A b2 A2
m211 2 ,
8A2 αA2 ω02
3b2 A b4 A2
m213 2 ,
8A2 αA2 ω02
5b4 A b6 A2
m215 2 ,
8A2 αA2 ω02
7b6 A2
m217 2 ,
8A2 αA2 ω02
m222 2b0 A 2b2 A b4 A2
,
2 16 αA2 ω02
m224 b2 A 2b4 A b6 A2
,
2 8 αA2 ω02
17
18
Mathematical Problems in Engineering
3b4 A 2b6 A2
m226 2 ,
16 αA2 ω02
b6 A2
m228 2 ,
4 αA2 ω02
m231 2b0 A − b2 A
−b2 A2b0 A 3A2 αω02 A αA2 ω02 2db0 A/dA db2 A/dA
,
×
3 8A2 αA2 ω02
m233 b2 A − b4 A
−b2 A b4 A 3A2 α ω02 A αA2 ω02 db2 A/dA db4 A/dA
,
×
3 8A2 αA2 ω02
b4 A − b6 A
m235 3 8A2 αA2 ω02
−b4 A b6 A 3A2 α ω02 A αA2 ω02 db4 A/dA db6 A/dA
×
,
3 8A2 αA2 ω02
m237 b6 A −b6 A 3A2 α ω02 A αA2 ω02 db6 A/dA
,
3
8A2 αA2 ω02
m242 2db A 2db A db A 0
2
4
−2Aα2b0 A 2b2 A b4 A A2 α ω02
dA
dA
dA
A2b0 A − b4 A
× 3 ,
32 αA2 ω02
m244 db A 2db A db A 2
4
6
−2Aαb2 A 2b4 A b6 A A2 α ω02
dA
dA
dA
Ab2 A − b6 A
× 3 ,
32 αA2 ω02
m246 Ab4 A
−2Aαb4 A 2b6 A A2 α ω02 db4 A/dA 2db6 A/dA
,
×
3 32 αA2 ω02
m248
Ab6 A −2Aαb6 A A2 α ω02 db6 A/dA
.
3 32 αA2 ω02
A.1
Mathematical Problems in Engineering
The diffusion coefficients in 5.8 are as follows:
b11 b111 b112 ,
b22 b221 b222 ,
bij 0,
i/
j,
b111 b1111 S1 ωA b1113 S1 3ωA b1115 S1 5ωA b1117 S1 7ωA,
b221 b2211 S1 ωA b2213 S1 3ωA b2215 S1 5ωA b2217 S1 7ωA,
b112 b1122 S2 2ωA b1124 S2 4ωA b1126 S2 6ωA b1128 S2 8ωA,
b222 b2220 S2 0 b2222 S2 2ωA b2224 S2 4ωA b2226 S2 6ωA b2228 S2 8ωA,
b1111 πb2 A − 2b0 A2
2 ,
4 αA2 ω02
b1113 πb2 A − b4 A2
2 ,
4 αA2 ω02
b1115 πb4 A − b6 A2
2 ,
4 αA2 ω02
πb62 A
b1117 2 ,
4 αA2 ω02
b1122 πA2 b4 A − 2b0 A2
2 ,
16 αA2 ω02
b1124 πA2 b2 A − b6 A2
2 ,
16 αA2 ω02
πA2 b42 A
b1126 2 ,
16 αA2 ω02
πA2 b62 A
b1128 2 ,
16 αA2 ω02
π2b0 A b2 A2
b2211 2 ,
4A2 αA2 ω02
πb2 A b4 A2
b2213 2 ,
4A2 αA2 ω02
19
20
Mathematical Problems in Engineering
πb4 A b6 A2
b2215 2 ,
4A2 αA2 ω02
πb62 A
b2217 2 ,
4A2 αA2 ω02
b2220 π2b0 A b2 A2
2 ,
8 αA2 ω02
b2222 π2b0 A 2b2 A b4 A2
,
2 16 αA2 ω02
b2224 πb2 A 2b4 A b6 A2
,
2 16 αA2 ω02
b2226 πb4 A 2b6 A2
2 ,
16 αA2 ω02
πb2 A
b2228 6
2 .
16 αA2 ω02
A.2
Acknowledgments
The work reported in this paper was supported by the National Natural Science Foundation
of China under Grant nos. 10802030, 10902096 and Specialized Research Fund for Doctoral
Program of Higher Education of China under Grant no. 200802511005.
References
1 R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, USA, 1957.
2 C. S. Huang, S. Wang, and K. L. Teo, “Solving Hamilton-Jacobi-Bellman equations by a modified
method of characteristics,” Nonlinear Analysis, Theory, Methods and Applications, vol. 40, no. 1, pp. 279–
293, 2000.
3 W. Q. Zhu, “Nonlinear stochastic dynamics and control in Hamiltonian formulation,” Applied
Mechanics Reviews, vol. 59, no. 1–6, pp. 230–247, 2006.
4 H. J. Kushner and W. Runggaldier, “Nearly optimal state feedback control for stochastic systems with
wideband noise disturbances,” SIAM Journal on Control and Optimization, vol. 25, no. 2, pp. 298–315,
1987.
5 M. F. Dimentberg, D. V. Iourtchenko, and A. S. Bratus, “Optimal bounded control of steady-state
random vibrations,” Probabilistic Engineering Mechanics, vol. 15, no. 4, pp. 381–386, 2000.
6 J. H. Park and K. W. Min, “Bounded nonlinear stochastic control based on the probability distribution
for the sdof oscillator,” Journal of Sound and Vibration, vol. 281, no. 1-2, pp. 141–153, 2005.
7 W. Q. Zhu and M. L. Deng, “Optimal bounded control for minimizing the response of quasi-integrable
Hamiltonian systems,” International Journal of Non-Linear Mechanics, vol. 39, no. 9, pp. 1535–1546, 2004.
8 M. Deng and W. Zhu, “Feedback minimization of first-passage failure of quasi integrable Hamiltonian
systems,” Acta Mechanica Sinica/Lixue Xuebao, vol. 23, no. 4, pp. 437–444, 2007.
Mathematical Problems in Engineering
21
9 L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Reviews of Modern
Physics, vol. 70, no. 1, pp. 223–287, 1998.
10 N. Sri Namachchivaya, “Almost sure stability of dynamical systems under combined harmonic and
stochastic excitations,” Journal of Sound and Vibration, vol. 151, no. 1, pp. 77–90, 1991.
11 G. O. Cai and Y. K. Lin, “Nonlinearly damped systems under simultaneous broad-band and harmonic
excitations,” Nonlinear Dynamics, vol. 6, no. 2, pp. 163–177, 1994.
12 Z. L. Huang and W. Q. Zhu, “Exact stationary solutions of averaged equations of stochastically and
harmonically excited mdof quasi-linear systems with internal and/or external resonances,” Journal of
Sound and Vibration, vol. 204, no. 2, pp. 249–258, 1997.
13 Z. L. Huang, W. Q. Zhu, and Y. Suzuki, “Stochastic averaging of strongly non-linear oscillators under
combined harmonic and white-noise excitations,” Journal of Sound and Vibration, vol. 238, no. 2, pp.
233–256, 2000.
14 W. Q. Zhu and Y. J. Wu, “First-passage time of Duffing oscillator under combined harmonic and
white-noise excitations,” Nonlinear Dynamics, vol. 32, no. 3, pp. 291–305, 2003.
15 Z. L. Huang and W. Q. Zhu, “Stochastic averaging of quasi-integrable Hamiltonian systems under
combined harmonic and white noise excitations,” International Journal of Non-Linear Mechanics, vol.
39, no. 9, pp. 1421–1434, 2004.
16 Y. J. Wu and W. Q. Zhu, “Stochastic averaging of strongly nonlinear oscillators under combined
harmonic and wide-band noise excitations,” ASME Journal of Vibration and Acoustics, vol. 130, no.
5, Article ID 051004, 2008.
17 D. V. Iourtchenko, “Solution to a class of stochastic LQ problems with bounded control,” Automatica,
vol. 45, no. 6, pp. 1439–1442, 2009.
18 W. Q. Zhu and Y. J. Wu, “Optimal bounded control of first-passage failure of strongly non-linear
oscillators under combined harmonic and white-noise excitations,” Journal of Sound and Vibration, vol.
271, no. 1-2, pp. 83–101, 2004.
19 W. Q. Zhu and Y. J. Wu, “Optimal bounded control of harmonically and stochastically excited strongly
nonlinear oscillators,” Probabilistic Engineering Mechanics, vol. 20, no. 1, pp. 1–9, 2005.
20 Y. J. Wu and R. H. Huan, “First-passage failure minimization of stochastic Duffing-Rayleigh-Mathieu
system,” Mechanics Research Communications, vol. 35, no. 7, pp. 447–453, 2008.
21 Z. Xu and Y. K. Cheung, “Averaging method using generalized harmonic functions for strongly nonlinear oscillators,” Journal of Sound and Vibration, vol. 174, no. 4, pp. 563–576, 1994.
22 R. Z. Khasminskii, “A limit theorem for solution of differential equations with random right hand
sides,” Theory of Probability and Its Application, vol. 11, no. 3, pp. 390–405, 1966.
23 R. L. Stratonovich, Topics in the Theory of Random Noise, vol. 1, Gordon and Breach, New York, NY,
USA, 1963.
24 H. J. Kushner, “Optimality conditions for the average cost per unit time problem with a diffusion
model,” SIAM Journal on Control and Optimization, vol. 16, no. 2, pp. 330–346, 1978.
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